We begin with by noticing that the system (REF ) has the following conserved quantity [1]}:
\({\cal H}(t)={2}\int _\mathbb {E}(u^2+u_x^2+\rho ^2)dx={2}\Vert u(t,\cdot )\Vert ^2_{H^1(\mathbb {E})}+{2}\Vert \rho (t,\cdot )\Vert ^2_{L^2(\mathbb {E})}.\) 
